Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 9, simplify the following expressions. $$ \frac{(a b)^{-x}}{a^{-x}+b^{-y}} $$

Step-by-Step Solution

Answer

The simplified expression is \(1 + b^{y-x}\)

Step 1: Express the Terms Using Positive Exponents

When an expression has negative exponents, it's often easier to handle them when they're expressed as positive exponents. The rule which should be utilized here is that \(x^{-n} = \frac{1}{x^{n}}\). By applying this, the expression simplifies to: \[ \frac{1}{(a^{x} b^{x})}\div \frac{1}{(a^{x} + b^{y})}\]

Step 2: Simplify by Division

Division can be reinterpreted as multiplication by the reciprocal of the divisor. So the previous expressions simplifies further to: \[ \frac{1}{(a^{x} b^{x})} * \frac{(a^{x} + b^{y})}{1}\] or \[(a^{x} + b^{y}) / (a^{x} b^{x})\]

Step 3: Use the Distributive Property

Now, the distributive property of multiplication over addition can be applied to the numerator to get: \[a^{x}b^{-x} + b^{y}b^{-x}\]

Step 4: Get the Final Simplified Expression

Finally, the addition of the exponents rule is applied which gives for the exponents: \(x + -x = 0\) and \(y + -x\) remains as it is. This yields the simplified final expression: \[1 + b^{y-x}\]

Key Concepts

Exponents Rules

When dealing with exponents, it's essential to understand the fundamental rules that govern how they function in mathematical expressions. Exponents can be thought of as a shorthand notation for repeated multiplication. For instance, \( a^2 \) means \( a \) is multiplied by itself. Some key rules to remember are:

• Product of Powers Rule: \( x^m \times x^n = x^{m+n} \)
• Quotient of Powers Rule: \( \frac{x^m}{x^n} = x^{m-n} \)
• Power of a Power Rule: \( (x^m)^n = x^{m\times n} \)
• Power of a Product Rule: \( (xy)^n = x^n y^n \)

Understanding these rules allows you to manipulate and simplify expressions efficiently, such as combining or separating terms based on their powers. In simplifying complex expressions, these rules help us rewrite them in a more digestible form.

Distributive Property

The distributive property is a crucial tool for simplifying algebraic expressions that involve multiplication over addition or subtraction. It allows us to distribute, or "pass out," a factor across terms inside parentheses. The property is formally written as \( a(b + c) = ab + ac \). This property is exceptionally useful when:

• Dealing with polynomials or expressions with multiple terms in parentheses.
• Distributing terms to simplify equations or expressions for easier calculation.

In our specific exercise, the distributive property is used to manipulate terms in the numerator, combining and simplifying them effectively. By understanding and applying this property, you can break down complex expressions into more manageable parts.

Negative Exponents

Negative exponents represent reciprocal functions. For any non-zero number \( x \), \( x^{-n} \) is equivalent to \( \frac{1}{x^n} \). This concept is pivotal when simplifying expressions that initially involve negative exponents. In our exercise, converting negative exponents to positive ones makes the expression easier to handle:

• Negative exponents become positive by taking the reciprocal.
• This transformation simplifies division and multiplication operations.

When simplified to positive exponents, expressions often reveal a clearer structure, as shown in the step-by-step solution, where the expression \((ab)^{-x}\) was rewritten for better manageability.

Algebraic Manipulation

Algebraic manipulation involves the strategic use of algebraic principles to simplify or rearrange an expression. This process combines various techniques such as factoring, expanding, or rearranging parts of an equation. Key strategies include:

• Applying exponent rules to combine or divide terms.
• Using the distributive property to expand expressions effectively.
• Simplifying by recognizing patterns or similar structures within equations.

In the given exercise, algebraic manipulation was crucial to simplify the expression to \( 1 + b^{y-x} \). Through step-by-step simplification, we applied exponent rules and the distributive property to ultimately condense the expression into its final form. Mastery of these techniques is foundational to solving complex algebraic problems efficiently.